。
My mother has a bad cold.
Literal
Mother [topic-は] terrible cold [to-に] caught-[ongoing-ている]
ひどい風邪にかかっている = 'has caught a terrible cold (and is still suffering).' ひどい = terrible/severe. ~にかかる = to catch (a disease). ている marks the ongoing state.
。
My mother has a bad cold.